A tail set crupper is a portion of a harness for an animal, such as a horse, that is used to encourage the tail to be maintained in an arched shape. It may be worn by the animal for extended periods of time. After a period of use, the crupper may tend to cause the tail to be straightened relative to the direction of the spine of the animal, and also to extend on a relatively upstanding, arcuate shape, such that the tail may take on the desired “set”, hence the “tail set” crupper. A horse, or other animal, may find the crupper uncomfortable. Consequently, the horse may make repeated efforts to dislodge the crupper. In the view of the present inventor, horses often are, or become, quite adept at dislodging the crupper. It might therefore be advantageous to have a crupper that might tend either to be more comfortable, or less easily dislodged, or both.